3 Non-linear Dynamics in Accelerators
75
We can define powers as:
(: f :)
2 g =: f : (: f : g) = [f, [f, g]] etc.
(3.74)
One can collect a set of useful formulae for calculations:
Some common special (very useful) cases for f :
: x : =
∂
∂p
: p : = −
∂
∂x
: x : 2 =
applied twice
: x :: x : =
∂ 2
∂p 2
: p : 2 =
applied twice
: p :: p : =
∂ 2
∂x 2
: xp : = p
∂
∂p − x
∂
∂x
: x :: p : = : p :: x : = −
∂ 2
∂x∂p
: x 2 : = 2x
∂
∂p
: p 2 : = − 2p
∂
∂x
: x n : = n · x n−1 ∂
∂p
: p n : = − n · p n−1 ∂
∂x
(3.75)
Once powers of the Lie operators are defined, they can be used to formulated an
exponential form:
e
:f :
=
∞
i=0
1
i!
(: f :)
i
(3.76)
This expression is call a “Lie transformation”.
Give the Hamiltonian H of an element, the generator f is this Hamiltonian
multiplied by the length L of the element.
To evaluate a simple example, for the case H = −p 2 /2 using the exponential
form and (3.75):
e : −Lp 2 /2 : x = x −
1
2
L : p
2
: x +
1
8
L
2 (: p
2
:)
2 x + ..
= x + Lp
(3.77)
e : −Lp 2 /2 : p = p −
1
2
L : p
2
: p + . . .
= p
(3.78)
One can easily verify that for 1D and δ = 0 this is the transformation of a drift space
of length L (if p ≈ x ) as introduced previously. The function f (x, p) = −Lp 2 /2
is the generator of this transformation.